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A Fr i en
o
troduction n I y t dl
Mathematical
Logic 2nd Edition
Christopher C. Leary Lars Kristiansen
A Friendly Introduction to Mathematical Logic
A Friendly Introduction to Mathematical Logic 2nd Edition
Christopher C. Leary State University of New York College at Geneseo
Lars Kristiansen The University of Oslo
Milne Library, SUNY Geneseo, Geneseo, NY
c
2015 Christopher C. Leary and Lars Kristiansen ISBN: 978-1-942341-07-9 (paperback) ISBN: 978-1-942341-32-1 (ebook)
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. You are free to: Share—copy and redistribute the material in any medium or format Adapt—remix, transform, and build upon the material The licensor cannot revoke these freedoms as long as you follow the license terms. Under the following terms: Attribution—You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. NonCommercial—You may not use the material for commercial purposes. ShareAlike—If you remix, transform, or build upon the material, you must distribute your contributions under the same license as the original.
Milne Library SUNY Geneseo One College Circle Geneseo, NY 14454 Lars Kristiansen has received financial support from the Norwegian Non-fiction Literature Fund
viii 8 Summing Up, Looking Ahead 8.1 Once More, With Feeling . . . . . . . . . 8.2 The Language LBT and the Structure B. 8.3 Nonstandard LBT -structures . . . . . . . 8.4 The Axioms of B . . . . . . . . . . . . . . 8.5 B extended with an induction scheme . . 8.6 Incompleteness . . . . . . . . . . . . . . . 8.7 Off You Go . . . . . . . . . . . . . . . . .
CONTENTS
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265 266 266 271 271 274 276 278
Appendix: Just Enough Set Theory to Be Dangerous
279
Solutions to Selected Exercises
283
Bibliography
359
Preface Preface to the First Edition This book covers the central topics of first-order mathematical logic in a way that can reasonably be completed in a single semester. From the core ideas of languages, structures, and deductions we move on to prove the Soundness and Completeness Theorems, the Compactness Theorem, and G¨odel’s First and Second Incompleteness Theorems. There is an introduction to some topics in model theory along the way, but I have tried to keep the text tightly focused. One choice that I have made in my presentation has been to start right in on the predicate logic, without discussing propositional logic first. I present the material in this way as I believe that it frees up time later in the course to be spent on more abstract and difficult topics. It has been my experience in teaching from preliminary versions of this book that students have responded well to this choice. Students have seen truth tables before, and what is lost in not seeing a discussion of the completeness of the propositional logic is more than compensated for in the extra time for G¨ odel’s Theorem. I believe that most of the topics I cover really deserve to be in a first course in mathematical logic. Some will question my inclusion of the L¨ owenheim–Skolem Theorems, and I freely admit that they are included mostly because I think they are so neat. If time presses you, that section might be omitted. You may also want to soft-pedal some of the more technical results in Chapter 5. The list of topics that I have slighted or omitted from the book is depressingly large. I do not say enough about recursion theory or model theory. I say nothing about linear logic or modal logic or second-order logic. All of these topics are interesting and important, but I believe that they are best left to other courses. One semester is, I believe, enough time to cover the material outlined in this book relatively thoroughly and at a reasonable pace for the student. Thanks for choosing my book. I would love to hear how it works for you.
ix
x
Preface
To the Student Welcome! I am really thrilled that you are interested in mathematical logic and that we will be looking at it together! I hope that my book will serve you well and will help to introduce you to an area of mathematics that I have found fascinating and rewarding. Mathematical logic is absolutely central to mathematics, philosophy, and advanced computer science. The concepts that we discuss in this book—models and structures, completeness and incompleteness—are used by mathematicians in every branch of the subject. Furthermore, logic provides a link between mathematics and philosophy, and between mathematics and theoretical computer science. It is a subject with increasing applications and of great intrinsic interest. One of the tasks that I set for myself as I wrote this book was to be mindful of the audience, so let me tell you the audience that I am trying to reach with this book: third- or fourth-year undergraduate students, most likely mathematics students. The student I have in mind may not have taken very many upper-division mathematics courses. He or she may have had a course in linear algebra, or perhaps a course in discrete mathematics. Neither of these courses is a prerequisite for understanding the material in this book, but some familiarity with proving things will be required. In fact, you don’t need to know very much mathematics at all to follow this text. So if you are a philosopher or a computer scientist, you should not find any of the core arguments beyond your grasp. You do, however, have to work abstractly on occasion. But that is hard for all of us. My suggestion is that when you are lost in a sea of abstraction, write down three examples and see if they can tell you what is going on. At several points in the text there are asides that are indented and start with the word Chaff. I hope you will find these comments helpful. They are designed to restate difficult points or emphasize important things that may get lost along the way. Sometimes they are there just to break up the exposition. But these asides really are chaff, in the sense that if they were blown away in the wind, the mathematics that is left would be correct and secure. But do look at them—they are supposed to make your life easier. Just like every other math text, there are exercises and problems for you to work out. Please try to at least think about the problems. Mathematics is a contact sport, and until you are writing things down and trying to use and apply the material you have been studying, you don’t really know the subject. I have tried to include problems of different levels of difficulty, so some will be almost trivial and others will give you a chance to show off. This is an elementary textbook, but elementary does not mean easy. It was not easy when we learned to add, or read, or write. You will find the going tough at times as we work our way through some very difficult and technical results. But the major theorems of the course—G¨odel’s Com-
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xi
pleteness Theorem, the incompleteness results of G¨odel and Rosser, the Compactness Theorem, the L¨ owenheim–Skolem Theorem—provide wonderful insights into the nature of our subject. What makes the study of mathematical logic worthwhile is that it exposes the core of our field. We see the strength and power of mathematics, as well as its limitations. The struggle is well worth it. Enjoy the ride and see the sights.
Thanks Writing a book like this is a daunting process, and this particular book would never have been produced without the help of many people. Among my many teachers and colleagues I would like to express my heartfelt thanks to Andreas Blass and Claude Laflamme for their careful readings of early versions of the book, for the many helpful suggestions they made, and for the many errors they caught. I am also indebted to Paul Bankston of Marquette University, William G. Farris of the University of Arizona at Tucson, and Jiping Liu of the University of Lethbridge for their efforts in reviewing the text. Their thoughtful comments and suggestions have made me look smarter and made my book much better. TheDepartment of Mathematics at SUNY Geneseo has been very supportive of my efforts, and I would also like to thank the many students at Oberlin and at Geneseo who have listened to me lecture about logic, who have challenged me and rewarded me as I have tried to bring this field alive for them. The chance to work with undergraduates was what brought me into this field, and they have never (well, hardly ever) disappointed me. Much of the writing of this book took place when I was on sabbatical during the fall semester of 1998. The Department of Mathematics and Statistics at the University of Calgary graciously hosted me during that time so I could concentrate on my writing. I would also like to thank Michael and Jim Henle. On September 10, 1975, Michael told a story in Math 13 about a barber who shaves every man in his town that doesn’t shave himself, and that story planted the seed of my interest in logic. Twenty-two years later, when I was speaking with Jim about my interest in possibly writing a textbook, he told me that he thought that I should approach my writing as a creative activity, and if the book was in me, it would come out well. His comment helped give me the confidence to dive into this project. The typesetting of this book depended upon the existence of Leslie Lamport’s LATEX. I thank everyone who has worked on this typesetting system over the years, and I owe a special debt to David M. Jones for his Index package, and to Piet von Oostrum for Fancyheadings. Many people at Prentice Hall have worked very hard to make this book a reality. In particular, George Lobell, Gale Epps, and Lynn Savino have
xii
Preface
been very helpful and caring. You would not be holding this book without their efforts. But most of all, I would like to thank my wife, Sharon, and my children, Heather and Eric. Writing this book has been like raising another child. But the real family and the real children mean so much more.
Preface to the Second Edition From Chris: I was very happy with the reaction to the first edition of A Friendly Introduction. I heard from many readers with comments, errors (both small and embarrassingly large), and requests for solutions to the exercises. The many kind words and thoughtful comments were and are much appreciated, and most, if not all, of your suggestions have been incorporated into the work you have before you. Thank you all! As is often the case in publishing ventures, after a while the people at Prentice-Hall thought that the volume of sales of my book was not worth it to them, so they took the book out of print and returned the rights to me. I was very pleased when I received an email from Lars Kristiansen in September of 2012 suggesting that we work together on a second edition of the text and with the idea of including a section on computability theory as well as solutions to some of the exercises, solutions that he had already written up. This has allowed us to chart two paths to the incompleteness theorems, splitting after the material in Chapter 4. Readers of the first edition will find that the exposition in Chapters 5 and 6 follows a familiar route, although the material there has been pretty thoroughly reworked. It is also possible, if you choose, to move directly from Chapter 4 to Chapter 7 and see a development of computability theory that covers the Entscheidungsproblem, Hilbert’s 10th Problem, and G¨odel’s First Incompleteness Theorem. I am more than happy to have had the chance to work with Lars on this project for the last couple of years, and to have had his careful and creative collaboration. Lars has added a great deal to the work and has improved it in many ways. I am also in debt to the Department of Mathematics at the University of Oslo for hosting me in Norway during a visit in 2013 so that Lars and I could work on the revision face-to-face. The staff at Milne Library of SUNY Geneseo have been most helpful and supportive as we have moved toward bringing this second edition to fruition. In particular, Cyril Oberlander, Katherine Pitcher, and Allison Brown have been encouraging and comforting as we have worked through the details of publication and production. As in the first edition, I mostly have to thank my family. Eric and Heather, you were two and five when the first edition came out. I don’t think either of you will read this book, even now, but I hope you know that
Preface
xiii
you are still my most important offspring. And Sharon, thanks to you for all of your support and love. Also thanks for taking one for the team and accompanying me to Oslo when I had to work with Lars. I know what a sacrifice that was. This edition of the book is much longer than the original, and I am confident that it is a whole lot better. But the focus of the book has not changed: Lars and I believe that we have outlined an introduction to important areas of mathematical logic, culminating in the Incompleteness Theorems, that can reasonably be covered in a one-semester upper division undergraduate course. We hope that you agree!
From Lars: First of all, I will say thank you to Chris for letting me in on this project. We have worked very well together and complemented each other in a number of respects. I should also express my thanks to those who through the years have shaped my academic taste and pursuits. They have in some sense contributed to this book. Among them you find my teachers, colleagues and students at the University of Oslo. I cannot mention them all – I can probably not even remember them all – but a few names that immediately come to my mind are St˚ al Aanderaa, Herman Ruge Jervell (my PhD supervisor), Dag Normann, and Mathias Barra. Finally, I will like to thank Dag Normann and Amir Ben-Amram for discussions and helpful comments on early versions of Chapter 7. Our target group is undergraduate students that have reached a certain level of mathematical maturity but do not know much formal logic – maybe just some propositional logic – maybe nothing. It is the needs of the readers in this group that we want to meet, and we have made our efforts to do so: We have provided exercises of all degrees of difficulty, and we have provided detailed solutions to quite a few of them. We have provided discussions and explanations that might prevent unnecessary misunderstandings. We have stuck to topics that should be of interest to the majority of our target group. We have tried to motivate our definitions and theorems . . . and we have done a number of other things that hopefully will help an undergraduate student that wants to learn mathematical logic. This book conveys some of the main insights from what we today call classic mathematical logic. We tend to associate the word “classic” with something old. But the theorems in this book are not old. Not if we think about the pyramids. Neither if we think about Pythagoras, Euclid, and Diophantus – or even Newton and Leibniz. All the theorems in this book were conceived after my grandparents were born, some of them even after I was born. They are insights won by the past few generations. Many things that seem very important to us today will be more or less forgotten in a hundred years or so. The essence of classic mathematical logic will be
xiv
Preface
passed on from generation to generation as long as the human civilization exists. So, in some sense, this is a book for the future. I dedicate this book to the coming generations and, in particular, to my seven-year-old daughter Mille.
Chapter 1
Structures and Languages Let us set the stage. In the middle of the nineteenth century, questions concerning the foundations of mathematics began to appear. Motivated by developments in geometry and in calculus, and pushed forward by results in set theory, mathematicians and logicians tried to create a system of axioms for mathematics, in particular, arithmetic. As systems were proposed, notably by the German mathematician Gottlob Frege, errors and paradoxes were discovered. So other systems were advanced. At the International Congress of Mathematicians, a meeting held in Paris in 1900, David Hilbert proposed a list of 23 problems that the mathematical community should attempt to solve in the upcoming century. In stating the second of his problems, Hilbert said: But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms [of arithmetic]: To prove that they are not contradictory, that is, that a finite number of logical steps based upon them can never lead to contradictory results. (Quoted in [Feferman 98]) In other words, Hilbert challenged mathematicians to come up with a set of axioms for arithmetic that were guaranteed to be consistent, guaranteed to be paradox-free. In the first two decades of the twentieth century, three major schools of mathematical philosophy developed. The Platonists held that mathematical objects had an existence independent of human thought, and thus the job of mathematicians was to discover the truths about these mathematical objects. Intuitionists, led by the Dutch mathematician L. E. J. Brouwer, 1
2
Chapter 1. Structures and Languages held that mathematics should be restricted to concrete operations performed on finite structures. Since vast areas of modern mathematics depended on using infinitary methods, Brouwer’s position implied that most of the mathematics of the previous 3000 years should be discarded until the results could be reproved using finitistic arguments. Hilbert was appalled at this suggestion and he became the leading exponent of the Formalist school, which held that mathematics was nothing more than the manipulation of meaningless symbols according to certain rules and that the consistency of such a system was nothing more than saying that the rules prohibited certain combinations of the symbols from occurring. Hilbert developed a plan to refute the Intuitionist position that most of mathematics was suspect. He proposed to prove, using finite methods that the Intuitionists would accept, that all of classical mathematics was consistent. By using finite methods in his consistency proof, Hilbert was sure that his proof would be accepted by Brouwer and his followers, and then the mathematical community would be able to return to what Hilbert considered the more important work of advancing mathematical knowledge. In the 1920s many mathematicians became actively involved in Hilbert’s project, and there were several partial results that seemed to indicate that Hilbert’s plan could be accomplished. Then came the shock. On Sunday, September 7, 1930, at the Conference on Epistemology of the Exact Sciences held in K¨onigsberg, Germany, a 24-year-old Austrian mathematician named Kurt G¨odel announced that he could show that there is a sentence such that the sentence is true but not provable in a formal system of classical mathematics. In 1931 G¨ odel published the proof of this claim along with the proof of his Second Incompleteness Theorem, which said that no consistent formal system of mathematics could prove its own consistency. Thus Hilbert’s program was impossible, and there would be no finitistic proof that the axioms of arithmetic were consistent. Mathematics, which had reigned for centuries as the embodiment of certainty, had lost that role. Thus we find ourselves in a situation where we cannot prove that mathematics is consistent. Although we believe in our hearts that mathematics is consistent, we know in our brains that we will not be able to prove that fact, unless we are wrong. For if we are wrong, mathematics is inconsistent. And (as we will see) if mathematics is inconsistent, then it can prove anything, including the statement which says that mathematics is consistent. So do we throw our hands in the air and give up the study
1.1. Na¨ıvely
3
of mathematics? Of course not! Mathematics is still useful, it is still beautiful, and it is still interesting. It is an intellectual challenge. It compels us to think about great ideas and difficult problems. It is a wonderful field of study, with rewards for us all. What we have learned from the developments of the nineteenth and twentieth centuries is that we must temper our hubris. Although we can still agree with Gauss, who said that, “Mathematics is the Queen of the Sciences. . . ” she no longer can claim to be a product of an immaculate conception. Our study of mathematical logic will take us to a point where we can understand the statement and the proof of G¨odel’s Incompleteness Theorems. On our way there, we will study formal languages, mathematical structures, and a certain deductive system. The type of thinking, the type of mathematics that we will do, may be unfamiliar to you, and it will probably be tough going at times. But the theorems that we will prove are among the most revolutionary mathematical results of the twentieth century. So your efforts will be well rewarded. Work hard. Have fun.
1.1
Na¨ıvely
Let us begin by talking informally about mathematical structures and mathematical languages. There is no doubt that you have worked with mathematical models in several previous mathematics courses, although in all likelihood it was not pointed out to you at the time. For example, if you have taken a course in linear algebra, you have some experience working with R2 , R3 , and Rn as examples of vector spaces. In high school geometry you learned that the plane is a “model” of Euclid’s axioms for geometry. Perhaps you have taken a class in abstract algebra, where you saw several examples of groups: The integers under addition, permutation groups, and the group of invertible n × n matrices with the operation of matrix multiplication are all examples of groups—they are “models” of the group axioms. All of these are mathematical models, or structures. Different structures are used for different purposes. Suppose we think about a particular mathematical structure, for example R3 , the collection of ordered triples of real numbers. If we try to do plane Euclidean geometry in R3 , we fail miserably, as (for example) the parallel postulate is false in this structure. On the other hand, if we want to do linear algebra in R3 , all is well and good, as we can think of the points of R3 as vectors and let the scalars be real numbers. Then the axioms for a real vector space are all true when interpreted in R3 . We will say that R3
4
Chapter 1. Structures and Languages
is a model of the axioms for a vector space, whereas it is not a model for Euclid’s axioms for geometry. As you have no doubt noticed, our discussion has introduced two separate types of things to worry about. First, there are the mathematical models, which you can think of as the mathematical worlds, or constructs. Examples of these include R3 , the collection of polynomials of degree 17, the set of 3 × 2 matrices, and the real line. We have also been talking about the axioms of geometry and vector spaces, and these are something different. Let us discuss those axioms for a moment. Just for the purposes of illustration, let us look at some of the axioms which state that V is a real vector space. They are listed here both informally and in a more formal language: Vector addition is commutative: (∀u ∈ V )(∀v ∈ V )u + v = v + u. There is a zero vector: (∃0 ∈ V )(∀v ∈ V )v + 0 = v. One times anything is itself: (∀v ∈ V )1v = v. Don’t worry if the formal language is not familiar to you at this point; it suffices to notice that there is a formal language. But do let us point out a few things that you probably accepted without question. The addition sign that is in the first two axioms is not the same plus sign that you were using when you learned to add in first grade. Or rather, it is the same sign, but you interpret that sign differently. If the vector space under consideration is R3 , you know that as far as the first two axioms up there are concerned, addition is vector addition. Similarly, the 0 in the second axiom is not the real number 0; rather, it is the zero vector. Also, the multiplication in the third axiom that is indicated by the juxtaposition of the 1 and the v is the scalar multiplication of the vector space, not the multiplication of third grade. So it seems that we have to be able to look at some symbols in a particular formal language and then take those symbols and relate them in some way to a mathematical structure. Different interpretations of the symbols will lead to different conclusions as regards the truth of the formal statement. For example, if we take the commutivity axiom above and work with the space V being R3 but interpret the sign + as standing for cross product instead of vector addition, we see that the axiom is no longer true, as cross product is not commutative. These, then, are our next objectives: to introduce formal languages, to give an official definition of a mathematical structure, and to discuss truth in those structures. Beauty will come later.
1.2. Languages
1.2
5
Languages
We will be constructing a very restricted formal language, and our goal in constructing that language will be to be able to form certain statements about certain kinds of mathematical structures. For our work, it will be necessary to be able to talk about constants, functions, and relations, and so we will need symbols to represent them. Chaff: Let us emphasize this once more. Right now we are discussing the syntax of our language, the marks on the paper. We are not going to worry about the semantics, or meaning, of those marks until later—at least not formally. But it is silly to pretend that the intended meanings do not drive our choice of symbols and the way in which we use them. If we want to discuss left-hemi-semi-demi-rings, our formal language should include the function and relation symbols that mathematicians in this lucrative and exciting field customarily use, not the symbols involved in chess, bridge, or right-hemi-semi-para-fields. It is not our goal to confuse anyone more than is necessary. So you should probably go through the exercise right now of taking a guess at a reasonable language to use if our intended field of discussion was, say, the theory of the natural numbers. See Exercise 1. Definition 1.2.1. A first-order language L is an infinite collection of distinct symbols, no one of which is properly contained in another, separated into the following categories: 1. Parentheses: ( , ). 2. Connectives: ∨, ¬. 3. Quantifier: ∀. 4. Variables, one for each positive integer n: v1 , v2 , . . . , vn , . . . . The set of variable symbols will be denoted Vars. 5. Equality symbol: =. 6. Constant symbols: Some set of zero or more symbols. 7. Function symbols: For each positive integer n, some set of zero or more n-ary function symbols. 8. Relation symbols: For each positive integer n, some set of zero or more n-ary relation symbols.
6
Chapter 1. Structures and Languages
To say that a function symbol is n-ary (or has arity n) means that it is intended to represent a function of n variables. For example, + has arity 2. Similarly, an n-ary relation symbol will be intended to represent a relation on n-tuples of objects. This will be made formal in Definition 1.6.1. To specify a language, all we have to do is determine which, if any, constant, function, and relation symbols we wish to use. Many authors, by the way, let the equality symbol be optional, or treat the equality symbol as an ordinary binary (i.e., 2-ary) relation symbol. We will assume that each language has the equality symbol, unless specifically noted. Chaff: We ought to add a word about the phrase “no one of which is properly contained in another,” which appears in this definition. We have been quite vague about the meaning of the word symbol , but you are supposed to be thinking about marks made on a piece of paper. We will be constructing sequences of symbols and trying to figure out what they mean in the next few pages, and by not letting one symbol be contained in another, we will find our job of interpreting sequences to be much easier. For example, suppose that our language contained both the constant symbol ♥ and the constant symbol ♥♥ (notice that the first symbol is properly contained in the second). If you were reading a sequence of symbols and ran across ♥♥, it would be impossible to decide if this was one symbol or a sequence of two symbols. By not allowing symbols to be contained in other symbols, this type of confusion is avoided, leaving the field open for other types of confusion to take its place. Example 1.2.2. Suppose that we were taking an abstract algebra course and we wanted to specify the language of groups. A group consists of a set and a binary operation that has certain properties. Among those properties is the existence of an identity element for the operation. Thus, we could decide that our language will contain one constant symbol for the identity element, one binary operation symbol, and no relation symbols. We would get LG is {0, +}, where 0 is the constant symbol and + is a binary function symbol. Or perhaps we would like to write our groups using the operation as multiplication. Then a reasonable choice could be LG is {1,−1 , ·}, which includes not only the constant symbol 1 and the binary function symbol ·, but also a unary (or 1-ary) function symbol −1 , which is designed to pick out the inverse of an element of the group. As you can see, there is a fair bit of choice involved in designing a language.
1.2. Languages
7
Example 1.2.3. The language of set theory is not very complicated at all. We will include one binary relation symbol, ∈, and that is all: LST is {∈}. The idea is that this symbol will be used to represent the elementhood relation, so the interpretation of the string x ∈ y will be that the set x is an element of the set y. You might be tempted to add other relation symbols, such as ⊂, or constant symbols, such as ∅, but it will be easier to define such symbols in terms of more primitive symbols. Not easier in terms of readability, but easier in terms of proving things about the language. In general, to specify a language we need to list the constant symbols, the function symbols, and the relation symbols. There can be infinitely many [in fact, uncountably many (cf. the Appendix)] of each. So, here is a specification of a language: a(f1 )
L is {c1 , c2 , . . . , f1
a(f2 )
, f2
a(R1 )
, . . . , R1
a(R2 )
, R2
, . . . }.
a(f )
Here, the ci ’s are the constant symbols, the fi i ’s are the function syma(R ) bols, and the Ri i ’s are the relation symbols. The superscripts on the function and relation symbols indicate the arity of the associated symbols, so a is a mapping that assigns a natural number to a string that begins with an f or an R, followed by a subscripted ordinal. Thus, an official function symbol might look like this: 223 f17 , which would say that the function that will be associated with the 17th function symbol is a function of 223 variables. Fortunately, such dreadful detail will rarely be needed. We will usually see only unary or binary function symbols and the arity of each symbol will be stated once. Then the authors will trust that the context will remind the patient reader of each symbol’s arity.
1.2.1
Exercises
1. Carefully write out the symbols that you would want to have in a language L that you intend to use to write statements of elementary algebra. Indicate which of the symbols are constant symbols, and the arity of the function and relation symbols that you choose. Now write out another language, M (i.e., another list of symbols) with the same number of constant symbols, function symbols, and relation symbols that you would not want to use for elementary algebra. Think about the value of good notation. 2. What are good examples of unary (1-ary) functions? Binary functions? Can you find natural examples of relations with arity 1, 2, 3, and 4? As
8
Chapter 1. Structures and Languages you think about this problem, stay mindful of the difference between the function and the function symbol, between the relation and the relation symbol.
3. In the town of Sneezblatt there are three eating establishments: McBurgers, Chez Fancy, and Sven’s Tandoori Palace. Think for a minute about statements that you might want to make about these restaurants, and then write out L, the formal language for your theory of restaurants. Have fun with this, but try to include both function and relation symbols in L. What interpretations are you planning for your symbols? 4. You have been put in charge of drawing up the schedule for a basketball league. This league involves eight teams, each of which must play each of the other seven teams exactly two times: once at home and once on the road. Think of a reasonable language for this situation. What constants would you need? Do you need any relation symbols? Function symbols? It would be nice if your finished schedule did not have any team playing two games on the same day. Can you think of a way to state this using the formal symbols that you have chosen? Can you express the sentence which states that each team plays every other team exactly two times? 5. Let’s work out a language for elementary trigonometry. To get you started, let us suggest that you start off with lots of constant symbols— one for each real number. It is tempting to use the symbol 7 to stand for the number seven, but this runs into problems. (Do you see why this is illegal? 7, 77, 7/3, . . . .) Now, what functions would you like to discuss? Think of symbols for them. What are the arities of your function symbols? Do not forget that you need symbols for addition and multiplication! What relation symbols would you like to use? 6. A computer language is another example of a language. For example, the symbol := might be a binary function symbol, where the interpretation of the instruction x := 7 would be to alter the internal state of the computer by placing the value 7 into the position in memory referenced by the variable x. Think about the function associated with the binary function symbol if
, then
.
What are the inputs into this function? What sort of thing does the function do? Look at the statement If x + y > 3, then z := 7. Identify the function symbols, constant symbols, and relation symbols. What are the arities of each function and relation symbol?
1.3. Terms and Formulas
9
7. What would be a good language for the theory of vector spaces? This problem is slightly more difficult, as there are two different varieties of objects, scalars and vectors, and you have to be able to tell them apart. Write out the axioms of vector spaces in your language. Or, better yet, use a language that includes a unary function symbol for each real number so that scalars don’t exist as objects at all! 8. It is not actually necessary to include function symbols in the language, since a function is just a special kind of relation. Just to see an example, think about the function f : N → N defined by f (x) = x2 . Remembering that a relation on N × N is just a set of ordered pairs of natural numbers, find a relation R on N × N such that (x, y) is an element of R if and only if y = f (x). Convince yourself that you could do the same for any function defined on any domain. What condition must be true if a relation R on A × B is to be a function mapping A to B?
1.3
Terms and Formulas
Suppose that L is the language {0, +, <}, and we are going to use L to discuss portions of arithmetic. If we were to write down the string of symbols from L, (v1 + 0) < v1 , and the string v17 )(∀ + +(((0, you would probably agree that the first string conveyed some meaning, even if that meaning were incorrect, while the second string was meaningless. It is our goal in this section to carefully define which strings of symbols of L we will use. In other words, we will select the strings that will have meaning. Now, the point of having a language is to be able to make statements about certain kinds of mathematical systems. Thus, we will want the statements in our language to have the ability to refer to objects in the mathematical structures under consideration. So we will need some of the strings in our language to refer to those objects. Those strings are called the terms of L. Definition 1.3.1. If L is a language, a term of L is a nonempty finite string t of symbols from L such that either: 1. t is a variable, or 2. t is a constant symbol, or 3. t :≡ f t1 t2 . . . tn , where f is an n-ary function symbol of L and each of the ti is a term of L.
10
Chapter 1. Structures and Languages
A couple of things about this definition need to be pointed out. First, there is the symbol :≡ in the third clause. The symbol :≡ is not a part of the language L. Rather it is a meta-linguistic symbol that means that the strings of L-symbols on each side of the :≡ are identical. Probably the best natural way to read clause 3. would be to say that “t is f t1 t2 . . . tn .” The other thing to notice about Definition 1.3.1 is that this is a definition by recursion, since in the third clause of the definition, t is a term if it contains substrings that are terms. Since the substrings of t are shorter (contain fewer symbols) than t, and as none of the symbols of L are made up of other symbols of L, this causes no problems. Example 1.3.2. Let L be the language {0, 1, 2, . . . , +, ·}, with one constant symbol for each natural number and two binary function symbols. Here are some of the terms of L: 714, +3 2, · + 3 2 4. Notice that 1 2 3 is not a term of L, but rather is a sequence of three terms in a row. Chaff: The term +3 2 looks pretty annoying at this point, but we will use this sort of notation (called Polish notation) for functions rather than the infix notation (3 + 2) that you are used to. We are not really being that odd here: You have certainly seen some functions written in Polish notation: sin(x) and f (x, y, z) come to mind. We are just being consistent in treating addition in the same way. What makes it difficult is that it is hard to remember that addition really is just another function of two variables. But we are sure that by the end of this book, you will be very comfortable with that idea and with the notation that we are using. A couple of points are probably worth emphasizing, just this once. Notice that in the application of the function symbols, there are no parentheses and no commas. Also notice that all of our functions are written with the operator on the left. So instead of 3 + 2, we write +3 2. The reason for this is for consistency and to make sure that we can parse our expressions. Let us give an example. Suppose that, in some language or other, we R wrote down the string of symbols ♥U ↑ ♦## . Assume that two of our colleagues, Humphrey and Ingrid, were waiting in the hall while we wrote down the string. If Humphrey came into the room and announced that our string was a 3-ary function symbol followed by three terms, whereas Ingrid proclaimed that the string was really a 4-ary relation symbol followed by two terms, this would be rather confusing. It would be really confusing if they were both correct! So we need to make sure that the strings that we write down can be interpreted in only one way. This property, called unique readability, is addressed in Exercise 7 of Section 1.4.1. Chaff: Unique readability is one of those things that, in the opinion of the authors, is important to know, interesting to
1.3. Terms and Formulas
11
prove, and boring to read. Thus the proof is placed in (we do not mean “relegated to”) the exercises. Suppose that we look more carefully at the term · + 3 2 4. Assume for now that the symbols in this term are supposed to be interpreted in the usual way, so that · means multiply, + means add, and 3 means three. Then if we add some parentheses to the term in order to clarify its meaning, we get ·(+3 2) 4, which ought to have the same meaning as ·5 4, which is 20, just as you suspected. Rest assured that we will continue to use infix notation, commas, and parentheses as seem warranted to increase the readability (by humans) of this text. So f t1 t2 . . . tn will be written f (t1 , t2 , . . . , tn ) and +3 2 will be written 3 + 2, with the understanding that this is shorthand and that our official version is the version given in Definition 1.3.1. The terms of L play the role of the nouns of the language. To make meaningful mathematical statements about some mathematical structure, we will want to be able to make assertions about the objects of the structure. These assertions will be the formulas of L. Definition 1.3.3. If L is a first-order language, a formula of L is a nonempty finite string φ of symbols from L such that either: 1. φ :≡ = t1 t2 , where t1 and t2 are terms of L, or 2. φ :≡ Rt1 t2 . . . tn , where R is an n-ary relation symbol of L and t1 , t2 , . . . , tn are all terms of L, or 3. φ :≡ (¬α), where α is a formula of L, or 4. φ :≡ (α ∨ β), where α and β are formulas of L, or 5. φ :≡ (∀v)(α), where v is a variable and α is a formula of L. If a formula ψ contains the subformula (∀v)(α) [meaning that the string of symbols that constitute the formula (∀v)(α) is a substring of the string of symbols that make up ψ], we will say that the scope of the quantifier ∀ is α. Any symbol in α will be said to lie within the scope of the quantifier ∀. Notice that a formula ψ can have several different occurrences of the symbol ∀, and each occurrence of the quantifier will have its own scope. Also notice that one quantifier can lie within the scope of another. The atomic formulas of L are those formulas that satisfy clause (1) or (2) of Definition 1.3.3. You have undoubtedly noticed that there are no parentheses or commas in the atomic formulas, and you have probably decided that we will continue
12
Chapter 1. Structures and Languages
to use both commas and infix notation as seems appropriate. You are correct on both counts. So, instead of writing the official version < SSSSS0SS0 in a language containing constant symbol 0, unary function symbol S, and binary relation symbol <, we will write SSSSS0
<
SS0
or (after some preliminary definitions) 5 < 2. Also notice that we are using infix notation for the binary logical connective ∨. We hope that this will make your life somewhat easier. You will be asked in Exercise 8 in Section 1.4.1 to prove that unique readability holds for formulas as well as terms. We will, in our exposition, use different-size parentheses, different shapes of delimiters, and omit parentheses in order to improve readability without (we hope) introducing confusion on your part. Notice that a term is not a formula! If the terms are the nouns of the language, the formulas will be the statements. Statements can be either true or false. Nouns cannot. Much confusion can be avoided if you keep this simple dictum in mind. For example, suppose that you are looking at a string of symbols and you notice that the string does not contain either the symbol = or any other relation symbol from the language. Such a string cannot be a formula, as it makes no claim that can be true or false. The string might be a term, it might be nonsense, but it cannot be a formula. Chaff: We do hope that you have noticed that we are dealing only with the syntax of our language here. We have not mentioned that the symbol ¬ will be used for denial, or that ∨ will mean “or,” or even that ∀ means “for every.” Don’t worry, they will mean what you think they should mean. Similarly, do not worry about the fact that the definition of a formula left out symbols for conjunctions, implications, and biconditionals. We will get to them in good time.
1.3.1
Exercises
1. Suppose that the language L consists of two constant symbols, ♦ and ♥, a unary relation symbol U, a binary function symbol [, and a 3ary function symbol ]. Write down at least three distinct terms of the language L. Write down a couple of nonterms that look like they might be terms and explain why they are not terms. Write a couple of formulas and a couple of nonformulas that look like they ought to be formulas.
1.4. Induction
13
2. The fact that we write all of our operations on the left is important for unique readability. Suppose, for example, that we wrote our binary operations in the middle (and did not allow the use of parentheses). If our language included the binary function symbol #, then the term u#v#w could be interpreted two ways. This can make a difference: Suppose that the operation associated with the function symbol # is “subtract.” Find three real numbers u, v, and w such that the two different interpretations of u#v#w lead to different answers. Any nonassociative binary function will yield another counterexample to unique readability. Can you think of three such functions? 3. The language of number theory is LN T is {0, S, +, ·, E, <}, where the intended meanings of the symbols are as follows: 0 stands for the number zero, S is the successor function S(x) = x + 1, the symbols +, ·, and < mean what you expect, and E stands for exponentiation, so E(3, 2) = 9. Assume that LN T -formulas will be interpreted with respect to the nonnegative integers and write an LN T -formula to express the claim that p is a prime number. Can you write the statement of Lagrange’s Theorem, which states that every natural number is the sum of four squares? Write a formula stating that there is no largest prime number. How would we express the Goldbach Conjecture, that every even number greater than two can be expressed as the sum of two primes? What is the formal statement of the Twin Primes Conjecture, which says that there are infinitely many pairs (x, y) such that x and y are both prime and y = x + 2? The Bounded Gap Theorem, proven in 2013, says that there are infinitely many pairs of prime numbers that differ by 70,000,000 or less. Write a formal statement of that theorem. Use shorthand in your answers to this problem. For example, after you have found the formula which says that p is prime, call the formula Prime(p), and use Prime(p) in your later answers. 4. Suppose that our language has infinitely many constant symbols of the form 0 ,00 ,000 , . . . and no function or relation symbols other than =. Explain why this situation leads to problems by looking at the formula =000000 . Where in our definitions do we outlaw this sort of problem?
1.4
Induction
You are familiar, no doubt, with proofs by induction. They are the bane of most mathematics students from their first introduction in high school
14
Chapter 1. Structures and Languages
through the college years. It is our goal in this section to discuss the proofs by induction that you know so well, put them in a different light, and then generalize that notion of induction to a setting that will allow us to use induction to prove things about terms and formulas rather than just the natural numbers. Just to remind you of the general form of a proof by induction on the natural numbers, let us state and prove a familiar theorem, assuming for the moment that the set of natural numbers is {1, 2, 3, . . .}. Theorem 1.4.1. For every natural number n, 1 + 2 + ··· + n =
n(n + 1) . 2
Proof. If n = 1, simple computation shows that the equality holds. For the inductive case, fix k ≥ 1 and assume that 1 + 2 + ··· + k =
k(k + 1) . 2
If we add k + 1 to both sides of this equation, we get 1 + 2 + · · · + k + (k + 1) =
k(k + 1) + (k + 1), 2
and simplifying the right-hand side of this equation shows that (k + 1) (k + 1) + 1 , 1 + 2 + · · · + (k + 1) = 2 finishing the inductive step, and the proof. As you look at the proof of this theorem, you notice that there is a base case, when n = 1, and an inductive case. In the inductive step of the proof, we prove the implication If the formula holds for k, then the formula holds for k + 1. We prove this implication by assuming the antecedent, that the theorem holds for a (fixed, but unknown) number k, and from that assumption proving the consequent, that the theorem holds for the next number, k + 1. Notice that this is not the same as assuming the theorem that we are trying to prove. The theorem is a universal statement—it claims that a certain formula holds for every natural number. Looking at this from a slightly different angle, what we have done is to construct a set of numbers with a certain property. If we let S stand for the set of numbers for which our theorem holds, in our proof by induction we show the following facts about S:
1.4. Induction
15
1. The number 1 is an element of S. We prove this explicitly in the base case of the proof. 2. If the number k is an element of S, then the number k + 1 is an element of S. This is the content of the inductive step of the proof. But now, notice that we know that the collection of natural numbers can be defined as the smallest set such that: 1. The number 1 is a natural number. 2. If k is a natural number, then k + 1 is a natural number. So S, the collection of numbers for which the theorem holds, is identical with the set of natural numbers, thus the theorem holds for every natural number n, as needed. (If you caught the slight lie here, just substitute “superset” where appropriate.) So what makes a proof by induction work is the fact that the natural numbers can be defined recursively. There is a base case, consisting of the smallest natural number (“1 is a natural number”), and there is a recursive case, showing how to construct bigger natural numbers from smaller ones (“If k is a natural number, then k + 1 is a natural number”). Now, let us look at Definition 1.3.3, the definition of a formula. Notice that the five clauses of the definition can be separated into two groups. The first two clauses, the atomic formulas, are explicitly defined: For example, the first case says that anything that is of the form = t1 t2 is a formula if t1 and t2 are terms. These first two clauses form the base case of the definition. The last three clauses are the recursive case, showing how if α and β are formulas, they can be used to build more complex formulas, such as (α ∨ β) or (∀v)(α). Now since the collection of formulas is defined recursively, we can use an inductive-style proof when we want to prove that something is true about every formula. The inductive proof will consist of two parts, a base case and an inductive case. In the base case of the proof we will verify that the theorem is true about every atomic formula—about every string that is known to be a formula from the base case of the definition. In the inductive step of the proof, we assume that the theorem is true about simple formulas (α and β), and use that assumption to prove that the theorem holds a more complicated formula φ that is generated by a recursive clause of the definition. This method of proof is called induction on the complexity of the formula, or induction on the structure of the formula. There are (at least) two ways to think about the word “simple” in the last paragraph. One way in which a formula α might be simpler than a complicated formula φ is if α is a subformula of φ. The following theorem, although mildly interesting in its own right, is included here mostly so that you can see an example of a proof by induction in this setting:
16
Chapter 1. Structures and Languages
Theorem 1.4.2. Suppose that φ is a formula in the language L. Then the number of left parentheses occurring in φ is equal to the number of right parentheses occurring in φ. Proof. We will present this proof in a fair bit of detail, in order to emphasize the proof technique. As you become accustomed to proving theorems by induction on complexity, not so much detail is needed. Base Case. We begin our inductive proof with the base case, as you would expect. Our theorem makes an assertion about all formulas, and the simplest formulas are the atomic formulas. They constitute our base case. Suppose that φ is an atomic formula. There are two varieties of atomic formulas: Either φ begins with an equals sign followed by two terms, or φ begins with a relation symbol followed by several terms. As there are no parentheses in any term (we are using the official definition of term, here), there are no parentheses in φ. Thus, there are as many left parentheses as right parentheses in φ, and we have established the theorem if φ is an atomic formula. Inductive Case. The inductive step of a proof by induction on complexity of a formula takes the following form: Assume that φ is a formula by virtue of clause (3), (4), or (5) of Definition 1.3.3. Also assume that the statement of the theorem is true when applied to the formulas α and β. With those assumptions we will prove that the statement of the theorem is true when applied to the formula φ. Thus, as every formula is a formula either by virtue of being an atomic formula or by application of clause (3), (4), or (5) of the definition, we will have shown that the statement of the theorem is true when applied to any formula, which has been our goal. So, assume that α and β are formulas that contain equal numbers of left and right parentheses. Suppose that there are k left parentheses and k right parentheses in α and l left parentheses and l right parentheses in β. If φ is a formula by virtue of clause (3) of the definition, then φ :≡ (¬α). We observe that there are k + 1 left parentheses and k + 1 right parentheses in φ, and thus φ has an equal number of left and right parentheses, as needed. If φ is a formula because of clause (4), then φ :≡ (α ∨ β), and φ contains k + l + 1 left and right parentheses, an equal number of each type. Finally, if φ :≡ (∀v)(α), then φ contains k + 2 left parentheses and k + 2 right parentheses, as needed. This concludes the possibilities for the inductive case of the proof, so we have established that in every formula, the number of left parentheses is equal to the number of right parentheses. A second way in which we might structure a proof by induction on the structure of the formula is to say that α is simpler than φ if the number of connectives/quantifiers in α is less than the number in φ. In this case one
1.4. Induction
17
could argue that the induction argument is really an ordinary induction on the natural numbers. Here is an outline of how such a proof might proceed: Proof. We argue by induction on the structure of φ. Base Case. Assume φ has 0 connectives/quantifiers. This means that φ is an atomic formula. {Insert argument establishing the theorem for atomic formulas.} Inductive Case. Assume that φ has k + 1 connectives/quantifiers. Then either φ :≡ ¬α, or φ :≡ α ∨ β or φ :≡ (∀x)α, and we can assume that the theorem holds for every formula that has k or fewer connectives/quantifiers. We now argue that the theorem holds for the formula φ. {Insert arguments for the three inductive cases.} Between the base case and the inductive case we have established that the theorem holds for φ no matter how many connectives/quantifiers the formula φ contains, so by induction on the structure of φ, we have established that the theorem holds for all formulas φ. This might be a bit confusing on first glance, but the power of this proof technique will become very evident as you work through the following exercises and when we discuss the semantics of our language. Notice also that the definition of a term (Definition 1.3.1) is also a recursive definition, so we can use induction on the complexity of a term to prove that a theorem holds for every term.
1.4.1
Exercises
1. Prove, by ordinary induction on the natural numbers, that 12 + 22 + · · · + n2 =
n(n + 1)(2n + 1) . 6
2. Prove, by induction, that the sum of the interior angles in a convex n-gon is (n − 2)180◦ . (A convex n-gon is a polygon with n sides, where the interior angles are all less than 180◦ .) 3. Prove by induction that if A is a set consisting of n elements, then A has 2n subsets. 4. Suppose that L is {0, f, g}, where 0 is a constant symbol, f is a binary function symbol, and g is a 4-ary function symbol. Use induction on complexity to show that every L-term has an odd number of symbols. 5. If L is {0, <}, where 0 is a constant symbol and < is a binary relation symbol, show that the number of symbols in any formula is divisible by 3.
18
Chapter 1. Structures and Languages
6. If s and t are strings, we say that s is an initial segment of t if there is a nonempty string u such that t :≡ su, where su is the string s followed by the string u. For example, Kumq is an initial segment of Kumquat and +24 is an initial segment of +24u − v. Prove, by induction on the complexity of s, that if s and t are terms, then s is not an initial segment of t. [Suggestion: The base case, when s is either a variable or a constant symbol, should be easy. Then suppose that s is an initial segment of t and s :≡ f t1 t2 . . . tn , where you know that each ti is not an initial segment of any other term. Look for a contradiction.] 7. A language is said to satisfy unique readability for terms if, for each term t, t is in exactly one of the following categories: (a) Variable (b) Constant symbol (c) Complex term and furthermore, if t is a complex term, then there is a unique function symbol f and a unique sequence of terms t1 , t2 , . . . , tn such that t :≡ f t1 t2 . . . tn . Prove that our languages satisfy unique readability for terms. [Suggestion: You mostly have to worry about uniqueness—for example, suppose that t is c, a constant symbol. How do you know that t is not also a complex term? Suppose that t is f t1 t2 . . . tn . How do you show that the f and the ti ’s are unique? You may find Exercise 6 useful.] 8. To say that a language satisfies unique readability for formulas is to say that every formula φ is in exactly one of the following categories: (a) Equality (if φ :≡ = t1 t2 ) (b) Other atomic (if φ :≡ Rt1 t2 . . . tn for an n-ary relation symbol R) (c) Negation (d) Disjunction (e) Quantified Also, it must be that if φ is both = t1 t2 and = t3 t4 , then t1 is identical to t3 and t2 is identical to t4 , and similarly for other atomic formulas. Furthermore, if (for example) φ is a negation (¬α), then it must be the case that there is not another formula β such that φ is also (¬β), and similarly for disjunctions and quantified formulas. Prove that our languages satisfy unique readability for formulas. You will want to look at, and use, Exercise 7. You may have to prove an analog of Exercise 6, in which it may be helpful to think about the parentheses in an initial segment of a formula, in order to prove that no formula is an initial segment of another formula.
1.5. Sentences
19
9. Take the proof of Theorem 1.4.2 and write it out in the way that you would present it as part of a homework assignment. Thus, you should cut out all of the inessential motivation and present only what is needed to make the proof work.
1.5
Sentences
Among the formulas in the language L, there are some in which we will be especially interested. These are the sentences of L—the formulas that can be either true or false in a given mathematical model. Let us use an example to introduce a language that will be vitally important to us as we work through this book. Definition 1.5.1. The language LN T is {0, S, +, ·, E, <}, where 0 is a constant symbol, S is a unary function symbol, +, ·, and E are binary function symbols, and < is a binary relation symbol. This will be referred to as the language of number theory.
Chaff: Although we are not fixing the meanings of these symbols yet, we probably ought to tell you that the standard interpretation of LN T will use 0, +, ·, and < in the way that you expect. The symbol S will stand for the successor function that maps a number x to the number x + 1, and E will be used for exponentiation: E32 is supposed to be 32 . Consider the following two formulas of LN T : ¬(∀x)[(y < x) ∨ (y = x)]. (∀x)(∀y)[(x < y) ∨ (x = y) ∨ (y < x)]. (Did you notice that we have begun using an informal presentation of the formulas?) The second formula should look familiar. It is nothing more than the familiar trichotomy law of <, and you would agree that the second formula is a true statement about the collection of natural numbers, where you are interpreting < in the usual way. The first formula above is different. It “says” that not every x is greater than or equal to y. The truth of that statement is indeterminate: It depends on what natural number y represents. The formula might be true, or it might be false—it all depends on the value of y. So our goal in this section is to separate the formulas of L into one of two classes: the sentences (like the second example above) and the nonsentences. To begin this task, we must talk about free variables.
20
Chapter 1. Structures and Languages
Free variables are the variables upon which the truth value of a formula may depend. The variable y is free in the first formula above. To draw an analogy from calculus, if we look at Z x 1 dt, 1 t the variable x is free in this expression, as the value of the integral depends on the value of x. The variable t is not free, and in fact it doesn’t make any sense to decide on a value for t. The same distinction holds between free and nonfree variables in an L-formula. Let us try to make things a little more precise. Definition 1.5.2. Suppose that v is a variable and φ is a formula. We will say that v is free in φ if 1. φ is atomic and v occurs in (is a symbol in) φ, or 2. φ :≡ (¬α) and v is free in α, or 3. φ :≡ (α ∨ β) and v is free in at least one of α or β, or 4. φ :≡ (∀u)(α) and v is not u and v is free in α. Thus, if we look at the formula ∀v2 ¬(∀v3 )(v1 = S(v2 ) ∨ v3 = v2 ), the variable v1 is free whereas the variables v2 and v3 are not free. A slightly more complicated example is (∀v1 ∀v2 (v1 + v2 = 0)) ∨ v1 = S(0). In this formula, v1 is free whereas v2 is not free. Especially when a formula is presented informally, you must be careful about the scope of the quantifiers and the placement of parentheses. We will have occasion to use the informal notation ∀xφ(x). This will mean that φ is a formula and x is among the free variables of φ. If we then write φ(t), where t is an L-term, that will denote the formula obtained by taking φ and replacing each occurrence of the variable x with the term t. This will all be defined more formally and more precisely in Definition 1.8.2. Definition 1.5.3. A sentence in a language L is a formula of L that contains no free variables. For example, if a language contained the constant symbols 0, 1, and 2 and the binary function symbol +, then the following are sentences: 1+1 = 2 and (∀x)(x + 1 = x). You are probably convinced that the first of these is true and the second of these is false. In the next two sections we will see that you might be correct. But then again, you might not be.
1.5. Sentences
1.5.1
21
Exercises
1. For each of the following, find the free variables, if any, and decide if the given formula is a sentence. The language includes a binary function symbol +, a binary relation symbol <, and constant symbols 0 and 2. (a) (∀x)(∀y)(x + y = 2) (b) (x + y < x) ∨ (∀z)(z < 0) (c) ((∀y)(y < x)) ∨ ((∀x)(x < y)) 2. Explain precisely, using the definition of a free variable, how you know that the variable v2 is free in the formula (∀v1 )(¬(∀v5 )(v2 = v1 + v5 )). 3. In mathematics, we often see statements such as sin2 x + cos2 x = 1. Notice that this is not a sentence, as the variable x is free. But we all agree that this statement is true, given the usual interpretations of the symbols. How can we square this with the claim that sentences are the formulas that can be either true or false? 4. If we look at the first of our example formulas in this section, ¬(∀x)[(y < x) ∨ (y = x)], and we interpret the variables as ranging over the natural numbers, you will probably agree that the formula is false if y represents the natural number 0 and true if y represents any other number. (If you aren’t happy with 0 being a natural number, then use 1.) On the other hand, if we interpret the variables as ranging over the integers, what can we say about the truth or falsehood of this formula? Can you think of an interpretation for the symbols that would make sense if we try to apply this formula to the collection of complex numbers? 5. A variable may occur several times in a given formula. For example, the variable v1 occurs four times in the formula (∀v1 ) (v1 = v3 ) ∨ (v1 = Sv2 ) ∨ (0 + v17 < v1 − S0) . What should it mean for an occurrence of a variable to be free? Write a definition that begins: The nth occurrence of a variable v in a formula φ is said to be free if . . . . An occurrence of v in φ that is not free is said to be bound. Give an example of a formula in a suitable language that contains both free and bound occurrences of a variable v. 6. Look at the formula (∀y)(x = y) ∨ (∀x)(x < 0) .
22
Chapter 1. Structures and Languages If we denote this formula by φ(x) and t is the term S0, find φ(t). [Suggestion: The trick here is to see that there is a bit of a lie in the discussion of φ(t) in the text. Having completed Exercise 5, we can now say that we only replace the free occurrences of the variable x when we move from φ(x) to φ(t).]
1.6
Structures
Let us, by way of example, return to the language LN T of number theory. Recall that LN T is {0, S, +, ·, E, <}, where 0 is a constant symbol, S is a unary function symbol, +, ·, and E are binary function symbols, and < is a binary relation symbol. We now want to discuss the possible mathematical structures in which we can interpret these symbols, and thus the formulas and sentences of LN T . “But wait!” cries the incredulous reader. “You just said that this is the language of number theory, so certainly we already know what each of those symbols means.” It is certainly the case that you know an interpretation for these symbols. The point of this section is that there are many different possible interpretations for these symbols, and we want to be able to specify which of those interpretations we have in mind at any particular moment. Probably the interpretation you had in mind (what we will call the standard model for number theory) works with the set of natural numbers {0, 1, 2, 3, . . . }. The symbol 0 stands for the number 0. Chaff: Carefully, now! The symbol 0 is the mark on the paper, the numeral. The number 0 is the thing that the numeral 0 represents. The numeral is something that you can see. The number is something that you cannot see. The symbol S is a unary function symbol, and the function for which that symbol stands is the successor function that maps a number to the next larger natural number. The symbols +, ·, and E represent the functions of addition, multiplication, and exponentiation, and the symbol < will be used for the “less than” relation. But that is only one of the ways that we might choose to interpret those symbols. Another way to interpret all of those symbols would be to work with the numbers 0 and 1, interpreting the symbol 0 as the number 0, S as the function that maps 0 to 1 and 1 to 0, + as addition mod 2, · as multiplication mod 2, and (just for variety) E as the function with constant value 1. The symbol < can still stand for the relation “less than.” Or, if we were in a slightly more bizarre mood, we could work in a universe consisting of Beethoven, Picasso, and Ernie Banks, interpreting the symbol 0 as Picasso, S as the identity function, < as equality, and each
1.6. Structures
23
of the binary function symbols as the constant function with output Ernie Banks. The point is that there is nothing sacred about one mathematical structure as opposed to another. Without determining the structure under consideration, without deciding how we wish to interpret the symbols of the language, we have no way of talking about the truth or falsity of a sentence as trivial as (∀v1 )(v1 < S(v1 )). Definition 1.6.1. Fix a language L. An L-structure A is a nonempty set A, called the universe of A, together with: 1. For each constant symbol c of L, an element cA of A, 2. For each n-ary function symbol f of L, a function f A : An → A, and 3. For each n-ary relation symbol R of L, an n-ary relation RA on A (i.e., a subset of An ). Notice that the domain of the function f A is the set An , so f A is defined for all elements of An . Later in the text we will have occasion to discuss partial functions, those whose domain is a proper subset of An , but for now our functions are total functions, defined on all of the advertised domain. Chaff: The letter A is a German Fraktur capital A. We will also have occasion to use A’s friends, B and C. N will be used for a particular structure involving the natural numbers. The use of this typeface is traditional (which means this is the way we learned it). For your handwritten work, probably using capital script letters will be the best. Often, we will write a structure as an ordered k-tuple, like this: A A A A A = (A, cA 1 , c2 , f1 , R1 , R2 ).
As you can see, the notation is starting to get out of hand once again, and we will not hesitate to simplify and abbreviate when we believe that we can do so without confusion. So, when we are working in LN T , we will often talk about the standard structure N = (N, 0, S, +, ·, E, <), where the constants, functions, and relations do not get the superscripts they deserve, and the authors trust that you will interpret N as the collection {0, 1, 2, . . .} of natural numbers, the symbol 0 to stand for the number zero, + to stand for addition, S to stand for the successor function, and so on. By the way, if you are not used to thinking of 0 as a natural number, do not panic. Set theorists see 0 as the most natural of objects, so we tend to include it in N without thinking about it.
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Chapter 1. Structures and Languages x Oberon Titania Puck Bottom
